Theorem u5lemaa 586

Description: Lemma for relevance implication study.

Assertion

Ref	Expression
u5lemaa	((a →5 b) ∩ a) = (a ∩ b)

Proof of Theorem u5lemaa

Step	Hyp	Ref	Expression
1		df-i5 47	. . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
2	1	ran 71	. 2 ((a →5 b) ∩ a) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∩ a)
3		comanr1 446	. . . . 5 a C (a ∩ b)
4		comanr1 446	. . . . . 6 a⊥ C (a⊥ ∩ b)
5	4	comcom6 441	. . . . 5 a C (a⊥ ∩ b)
6	3, 5	com2or 465	. . . 4 a C ((a ∩ b) ∪ (a⊥ ∩ b))
7		comanr1 446	. . . . 5 a⊥ C (a⊥ ∩ b⊥ )
8	7	comcom6 441	. . . 4 a C (a⊥ ∩ b⊥ )
9	6, 8	fh1r 455	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∩ a) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) ∪ ((a⊥ ∩ b⊥ ) ∩ a))
10	3, 5	fh1r 455	. . . . . 6 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) = (((a ∩ b) ∩ a) ∪ ((a⊥ ∩ b) ∩ a))
11		an32 76	. . . . . . . . 9 ((a ∩ b) ∩ a) = ((a ∩ a) ∩ b)
12		anidm 103	. . . . . . . . . 10 (a ∩ a) = a
13	12	ran 71	. . . . . . . . 9 ((a ∩ a) ∩ b) = (a ∩ b)
14	11, 13	ax-r2 35	. . . . . . . 8 ((a ∩ b) ∩ a) = (a ∩ b)
15		an32 76	. . . . . . . . 9 ((a⊥ ∩ b) ∩ a) = ((a⊥ ∩ a) ∩ b)
16		ancom 68	. . . . . . . . . 10 ((a⊥ ∩ a) ∩ b) = (b ∩ (a⊥ ∩ a))
17		ancom 68	. . . . . . . . . . . . . 14 (a ∩ a⊥ ) = (a⊥ ∩ a)
18	17	ax-r1 34	. . . . . . . . . . . . 13 (a⊥ ∩ a) = (a ∩ a⊥ )
19		dff 93	. . . . . . . . . . . . . 14 0 = (a ∩ a⊥ )
20	19	ax-r1 34	. . . . . . . . . . . . 13 (a ∩ a⊥ ) = 0
21	18, 20	ax-r2 35	. . . . . . . . . . . 12 (a⊥ ∩ a) = 0
22	21	lan 70	. . . . . . . . . . 11 (b ∩ (a⊥ ∩ a)) = (b ∩ 0)
23		an0 100	. . . . . . . . . . 11 (b ∩ 0) = 0
24	22, 23	ax-r2 35	. . . . . . . . . 10 (b ∩ (a⊥ ∩ a)) = 0
25	16, 24	ax-r2 35	. . . . . . . . 9 ((a⊥ ∩ a) ∩ b) = 0
26	15, 25	ax-r2 35	. . . . . . . 8 ((a⊥ ∩ b) ∩ a) = 0
27	14, 26	2or 67	. . . . . . 7 (((a ∩ b) ∩ a) ∪ ((a⊥ ∩ b) ∩ a)) = ((a ∩ b) ∪ 0)
28		or0 94	. . . . . . 7 ((a ∩ b) ∪ 0) = (a ∩ b)
29	27, 28	ax-r2 35	. . . . . 6 (((a ∩ b) ∩ a) ∪ ((a⊥ ∩ b) ∩ a)) = (a ∩ b)
30	10, 29	ax-r2 35	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) = (a ∩ b)
31		ancom 68	. . . . 5 ((a⊥ ∩ b⊥ ) ∩ a) = (a ∩ (a⊥ ∩ b⊥ ))
32	30, 31	2or 67	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) ∪ ((a⊥ ∩ b⊥ ) ∩ a)) = ((a ∩ b) ∪ (a ∩ (a⊥ ∩ b⊥ )))
33	3, 8	fh4 454	. . . . 5 ((a ∩ b) ∪ (a ∩ (a⊥ ∩ b⊥ ))) = (((a ∩ b) ∪ a) ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
34		ax-a2 30	. . . . . . . 8 ((a ∩ b) ∪ a) = (a ∪ (a ∩ b))
35		a5b 112	. . . . . . . 8 (a ∪ (a ∩ b)) = a
36	34, 35	ax-r2 35	. . . . . . 7 ((a ∩ b) ∪ a) = a
37	36	ran 71	. . . . . 6 (((a ∩ b) ∪ a) ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
38	3, 8	fh1 451	. . . . . . 7 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b⊥ )))
39		anass 69	. . . . . . . . . . 11 ((a ∩ a) ∩ b) = (a ∩ (a ∩ b))
40	39	ax-r1 34	. . . . . . . . . 10 (a ∩ (a ∩ b)) = ((a ∩ a) ∩ b)
41	40, 13	ax-r2 35	. . . . . . . . 9 (a ∩ (a ∩ b)) = (a ∩ b)
42		anass 69	. . . . . . . . . . 11 ((a ∩ a⊥ ) ∩ b⊥ ) = (a ∩ (a⊥ ∩ b⊥ ))
43	42	ax-r1 34	. . . . . . . . . 10 (a ∩ (a⊥ ∩ b⊥ )) = ((a ∩ a⊥ ) ∩ b⊥ )
44		ancom 68	. . . . . . . . . . 11 ((a ∩ a⊥ ) ∩ b⊥ ) = (b⊥ ∩ (a ∩ a⊥ ))
45	19	lan 70	. . . . . . . . . . . . 13 (b⊥ ∩ 0) = (b⊥ ∩ (a ∩ a⊥ ))
46	45	ax-r1 34	. . . . . . . . . . . 12 (b⊥ ∩ (a ∩ a⊥ )) = (b⊥ ∩ 0)
47		an0 100	. . . . . . . . . . . 12 (b⊥ ∩ 0) = 0
48	46, 47	ax-r2 35	. . . . . . . . . . 11 (b⊥ ∩ (a ∩ a⊥ )) = 0
49	44, 48	ax-r2 35	. . . . . . . . . 10 ((a ∩ a⊥ ) ∩ b⊥ ) = 0
50	43, 49	ax-r2 35	. . . . . . . . 9 (a ∩ (a⊥ ∩ b⊥ )) = 0
51	41, 50	2or 67	. . . . . . . 8 ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b⊥ ))) = ((a ∩ b) ∪ 0)
52	51, 28	ax-r2 35	. . . . . . 7 ((a ∩ (a ∩ b)) ∪ (a ∩ (a⊥ ∩ b⊥ ))) = (a ∩ b)
53	38, 52	ax-r2 35	. . . . . 6 (a ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∩ b)
54	37, 53	ax-r2 35	. . . . 5 (((a ∩ b) ∪ a) ∩ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (a ∩ b)
55	33, 54	ax-r2 35	. . . 4 ((a ∩ b) ∪ (a ∩ (a⊥ ∩ b⊥ ))) = (a ∩ b)
56	32, 55	ax-r2 35	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∩ a) ∪ ((a⊥ ∩ b⊥ ) ∩ a)) = (a ∩ b)
57	9, 56	ax-r2 35	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∩ a) = (a ∩ b)
58	2, 57	ax-r2 35	1 ((a →5 b) ∩ a) = (a ∩ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 10   →₅ wi5 17

This theorem is referenced by:  u5lemnona 651  u5lembi 707

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i5 47  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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